Is there any way to limit reflection on a specific part of a mesh
Select the faces inside the ring and give them their own material. You can then assign them a material that doesn't reflect (or has diffuse reflections) and render that. If you really need to fake a reflective material inside the ring that can't see the rest of the ring....

Take that same new inner material and Set it to holdout. Do a render. Switch the material back to the same as before and do another render, but set the number of bounces in this new render to 1 or otherwise disable reflections. You can even hide the whole rest of the ring for the second render.  Then composite the two images together.